Problem: Simplify the following expression and state the condition under which the simplification is valid: $r = \dfrac{x^2 - 5x}{x^2 - 13x + 40}$
Answer: First factor the expressions in the numerator and denominator. $ \dfrac{x^2 - 5x}{x^2 - 13x + 40} = \dfrac{(x)(x - 5)}{(x - 8)(x - 5)} $ Notice that the term $(x - 5)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(x - 5)$ gives: $r = \dfrac{x}{x - 8}$ Since we divided by $(x - 5)$, $x \neq 5$. $r = \dfrac{x}{x - 8}; \space x \neq 5$